I can help you out here! It's not easy to wrap your head around at first because of how accustomed we are to the centralised internet by now, but it's actually fairly simple. Let's take Lemmy for example, anybody can create their own Lemmy instance (e.x. Beehaw.org, lemmy.world, lemmy.ml) and basically have their own functioning Reddit clone website. The fediverse/federated network only comes into play when a bunch of these instances connect to one another. You can totally just have a Lemmy instance and not federate with any other instances, but if you do, they share content with each other. This content is copied over between the two, and regularly synced (there's some more complicated wiring happening in the bg to make this more real-time, but what happens is essentially a sync between the content of the instances). In the case of Lemmy, that means the directory of subreddits on one instance is exposed to the other instance. This extends to the posts and comments as well. When you see a post from another server in your instance, it means that the instance you signed up on is "aware" of the other instance and shared content with it, and when you comment on that post, your comment is created in your instance and then later on shared with the other instance (which is why you can comment on a post in another instance and see it in that other instance).

That's how Lemmy takes advantage of the "fediverse" but it's not the only one. You've probably heard of Mastodon, which does the same except with tweets. There are also efforts to replicate other social media, such as YouTube (see: PeerTube), Facebook (see Friendica, GNU Social, and Misskey), Instagram (see PixelFed), and I'm sure others as well.

Tl;dr the fediverse is just a bunch of independently-run websites that share content with each other.